The state of the \(i\) th spin is updated from \(s_i^{(k-1)}\)  to \(s_i^{(k)}\)  if the energy is lowered, that is, \(\Delta E_i^{(k)} \le 0\) ; otherwise, the state is updated with probability \(\exp (- \Delta E_i^{(k)}/T_k)\) , where \(T_k\)  is the annealing schedule to lower the temperature.
That is, the new state \(s_i^{(k)}\)  is accepted with probability \(\min \lbrace 1, \exp (- \Delta E_i^{(k)}/T_k)\rbrace \) .
Annealing schedules used to lower the temperature often have \(T_k\)  that is proportional to \(1/k\)  or \( 1/ \log k\) .
See [1]}, [2]}, [3]}, and [4]}.
